332 M. F. August on a new Species 



and the sphere of which OA is the diameter. (This result 

 should be observed, since it can easily be deduced that the re- 

 flected ray forms a right angle with the cylinder, according to 

 the well-known property of the sphere.) 



The equation to the conical surface having this circle as a 

 base, and the point A as an apex, may be easily shown to be as 

 follows : 



(a? 2 -f if) cos 2 7 — zx cos a cos ft— zy cos ft cos y + zr sin 2 7 cos y 

 — rx cos 2 7 cos a — ry cos 2 7 cos 2 ft = 0. 



If, now, the eyes be placed symmetrically with respect to the 

 plane yz, so that both the lines drawn from the origin to the 

 eyes make the angle ft with the axis of y, and 7 with the axis 

 of z y and a and 180°— a respectively with the axis of x, then the 

 equation of the corresponding conical surface for the other eye 

 will be 



(a? 2 -f y 2 ) cos 2 y + zx cos a cos ft — zy cos ft cos 7 -f zr sin 2 7 cos 7 

 -f rx cos 2 7 cos a — ry cos 2 y cos ft = 0. 

 The subtraction of the two equations gives 



2zx cos a cos 7 + %rx cos 2 7 cos a = 0, 

 or 



2 cos a cos yx (z — r cos 7) = 0. 



The conical surfaces corresponding to the two eyes cut each 

 other, therefore, in two planes, (1) #=0, and (2) z=—r cos 7. 



The curve in the first plane is an ellipse, hyperbola, or para- 

 bola. (The case in which it becomes a circle cannot arise.) 



The curve in the second plane, z= —r cos 7,1s a circle, of which 

 the equation referred to rectangular coordinates is 



a7 2 -f-7/ 2 =r 2 sm 2 7. 



The particular case above considered may be very easily exhi- 

 bited experimentally ; the axis of rotation of the apparatus is 

 placed parallel to the incident rays, the eyes are put in such a 

 position that they are both equally distant from the origin and 

 Oz the axis of rotation. (Since y l = 0, the axis of y may be taken 

 anywhere in the axis of rotation without affecting the formula?.) 

 In this position of the eyes a stereoscopic image of a circle will 

 be observed as far behind the plane of rotation as the eyes are 

 before it, or conversely, as far before it as the eyes are behind, 

 having its middle point in the axis of rotation, and a radius equal 

 to the distance of either eye from the axis. It appears, therefore, 

 that it is the second only of the two curves mentioned above that 

 presents itself. Why this is the case shall be explained hereafter ; 

 our immediate purpose was only to verify our calculations by 

 comparison with a simple experiment. 



